This three nens system is fairly primitive and from Bankei's Zen perspective pointless.

It is primitive, as it shows how certain teachers know little to nothing about 'fundamental' teachings, Abhidharma and Yogacara, as both contain various explanations and systems for analysing mind and mind moments. (E.g. The Cittas) But even before when turn to complicated Buddhist scholastic texts, the five aggregates explain quite well what the "three nens" tries to say. The first nen of direct sensual experience is the body aggregate, the second nen of judging and naming is the second and third aggregate, the third nen of reflection third and fourth aggregate. Now, according to the teachings, the problem is not with the aggregates themselves, but with attachment and wrong identification. Clinging to only bodily impressions (first nen) is not only pointless (does not lead to liberation) but also just another form of self view.

Zen in general, and Bankei in particular is not about setting up systems to explain how the mind works. First of all, there are already texts and traditions that do just that. So, if one wants an in depth teaching, there are sources and teachers to turn to. Bankei taught about the Unborn, that is, a mind that is not bound to any state but open and aware. It is unborn, as it is not born as this or that - e.g. angry, happy, mindful, senseless, etc. - in other words, without self, without identification. To say that one should practise with or stay in a particular state of mind, that is exactly giving birth to a mind.

1Myriad dharmas are only mind.
Mind is unobtainable.
What is there to seek?2If the Buddha-Nature is seen,
there will be no seeing of a nature in any thing.3Neither cultivation nor seated meditation —
this is the pure Chan of Tathagata.4With sudden enlightenment to Tathagata Chan,
the six paramitas and myriad means
are complete within that essence.